For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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67 views

If matrix $M$ represents rotation around the origin, how to represent rotation about another point in terms of $M$?

For homework from school I have to made some tasks. There are no lessons because it is a second change. my question is: Multiplication by matrix $M$ represents rotation around the origin. If we do ...
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1answer
47 views

Stuck at Smith normal form

Can somebody help me with the Smith normal form of this matrix? I know what it should be, but I get stuck at some point. Can you show how to take it from the point I'm stuck? This is the matrix: ...
2
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2answers
109 views

Prove that the determinants are equal

$$ Let\ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
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1answer
62 views

covariance matrix is not positive definite

I have a feature vector(FV1) of size 1*n. Now I subtract mean of all feature vectors from the feature vector FV1 Now I take transpose of that(FV1_Transpose) which is n*1. Now I add do matrix ...
3
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2answers
121 views

Determine if a particular matrix is diagonalizable

my teacher gave me this exercise: Determine if this matrix is diagonalizable $ \begin{pmatrix} 1 & 1&1&1\\ 1&2&3&4\\ 1&-1&2&-2\\ 0&0&1&-2 ...
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0answers
181 views

What can be the possible rank of adjoint of matrix of order n? [duplicate]

Let $ A $ be matrix of order $ n $. What may the possible ranks of $\mathop{\rm adj} (A) $? I think the possible answers are $0$, $1$, and $n$.
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2answers
97 views

When is this matrix positive semi-definite?

I have a symmetric $n \times n$ matrix as follows. I want to find the eigenvalues of this Hessian matrix to state that it is not Positive Semi-Definite (i.e. some eigenvalues are negative while the ...
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0answers
25 views

$ || \lambda(A) - \lambda(B) ||_p \prec_k || \lambda(A -B) ||_p$?

Given two Hermitian matrices $\mathbf{A}$ and $\mathbf{B}$ and eigenvalue function $\lambda(\cdot)$ which returns eigenvalues of a matrix in non-increasing order. I found the following is true from ...
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1answer
129 views

2D boolean matrix number of unique combinations without mirrored/rotated ones

Given a $n \times n$ boolean matrix, it's well known that number of all possible combinations of 0s and 1s in that matrix would be $2^{n^2}$, as there are $n^2$ places which could take exactly 2 ...
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3answers
80 views

Does substitute $\lambda$ with matrix $A$ in a polynomial conflict with the Axiom of Substitution?

This seems to be an elementary question, gonna ask it anyway. Suppose that $A$ is a square matrix, and that $p(x)$ is its characteristic polynomial, we know that (1) $p(x) = \det(xE - A)$ We also ...
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1answer
53 views

Showing a matrix is nilpotent if its charateristic polynomial is $t^n$ mod ${\rm nil}(R)$

Let $R$ be a commutative ring. How to prove the following: If $\chi_A(t) \equiv t^n \bmod\operatorname{nil}(R)$ then $A \in M_n(R)$ is nilpotent. Note $\chi_A$ is the characteristic polynomial ...
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1answer
270 views

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
0
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2answers
212 views

Finding an Eigenvector of 3x3 matrix

I have a question $\lambda=4$ find an Eigenvector of a given 3x3 matrix. $ A = \left[ {\begin{array}{cc} 1 & 2 & 1 \\ 6 & 1& 0 \\ -1 & -2 & -1 \end{array} } \right] $ I know ...
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0answers
62 views

Generalization of trace norm identity

Given a $2\times 2$ complex matrix $M$, the sum of its singular values (i.e. the trace norm) can be written as: $$\mathrm{Tr}\,|M|=\sqrt{\mathrm{Tr}(M^\dagger M)+2|\mathrm{Det}(M)|}$$ Is anyone aware ...
0
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1answer
107 views

How can I refer a 3D pose (position + orientation) to a different coordinate system?

I'm working on a robotics project where all poses and marker positions/orientations are stored as a matrix: $$ \mathbf{P} =\begin{bmatrix} \mathbf{R} & \mathbf{t}\\ ...
2
votes
1answer
30 views

When is the matrix $A^{\ast} A$ isometric?

Are there conditions for a square matrix $A$ such that $A^{\ast} A$ is isometric, that is $\| A^{\ast} A x \| = \| x \|$ for all $x$?
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1answer
108 views

Is it possible to use the deflation algorithm to compute the eigenvalues of a large sparse matrix

I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully ...
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0answers
13 views

Sign of a Quadratic Form $X'AX + B'X+C$

We know that a quadratic form: $X'AX$, in the variable $X$, the matrices having their appropriate dimensions, is negative for all $X$, iff the matrix $A$ is negative definite. Can this result be ...
0
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1answer
72 views

Closed form solution to first order differential equation

I wonder why two solutions are not the same: The differential equation is: ds(t)/dt=A*s(t)+B(t) with initial condition s(0)=s0 A is a 2*2 constant matrix, s(t) is a 2*1 variable vector and B(t) ...
0
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1answer
221 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
3
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0answers
39 views

Solve non-linear equation with Matrices

I'm looking more for hints than specific answers, although I would be extremely grateful if provided with one. The problem I have is as follows: $$ -\Sigma (A+\Lambda_1)+I=0 $$ Here A is a constant, ...
2
votes
2answers
35 views

Proving a matrix identity: if $Z = VT^{-1}V^T$, then $ (I - Z + Z(I + Z)^{-1}Z)^{-1} = I + Z $

I'm walking through a least squares derivation for the Kalman Filter, and after several hours I'm still unable to derive the statement made on page 15. In particular, that for a matrix $Z = ...
5
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2answers
109 views

The periodic nature of the fibonacci sequence modulo $m$

Let $x_n$ denote the $n$-th element of the fibonacci sequence and $$A:=\begin{pmatrix} 0&1\\1&1 \end{pmatrix}$$ It's easy to show, that it holds: $$A^n=\begin{pmatrix} ...
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1answer
4k views

Finding a spanning set for a null space

I have a matrix $A$ that is like this: \begin{equation} A = \pmatrix{ 1 & 2 & -3 & 1 & 5 \\ 1 & 3 & -1 & 4 & -2 \\ 1 & 1 & -5 & -2 & 12 \\ 1 & 4 ...
0
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0answers
48 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
2
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0answers
45 views

Least number not being the determinant of a set of matrices

Let n > 1 be a natural number and u < v integers. How can I determine the least natural number not being the determinant of some n x n - matrix with integers in the range u..v without calculating ...
1
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3answers
62 views

Diagonalizability of a certain $4\times4$ matrix

Question $\bf 3.$ Determine if the following matrix is diagonalizable. (explain your answer) $$A=\pmatrix{ 1 & 4 & -2 & 3 \\ 3 & -3 & 0 & 4 \\ 1 & 1 & 1 ...
2
votes
0answers
43 views

Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
2
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1answer
81 views

Ask a question about an example in a course note on optimization problem with equality constraint

I have two difficulties on understanding the solution to an example in a course I took this semester on optimization. This example is given to illustrate the usage of Lagrange multiplier method ...
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1answer
38 views

True/false with justification. $A ∈ M_{n}(K)$, and for any $v ∈ M_{n×1}(K)$, $ Av = 0$, so $A = 0$.

Let $A ∈ M_n(K)$. If for all $v ∈ M_{n×1}(K)$, $Av = 0$, then $A = 0$. True or false? Mark scheme states Yes because for all $ 1 ≤ i ≤ n$, we take $$v = ...
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1answer
39 views

$SL (2, K)$ matrix conjugated by $GL(2,K)$

How can any arbitrary matrix is $SL_2 (K)$, where $K$ is any field be conjugated by some $GL_2 (K)$ element to $$ \begin{pmatrix} 0 & -1\\ 1 & x \end{pmatrix}? $$ Apologies in advance if the ...
0
votes
1answer
96 views

squaring the product of two anticommutative matrices

This is a very quick and simple question but I just need clarification. If I have the two matrices A and B that anti commute i.e. $AB=-BA$ does this mean that $(AB)^2=(BA)^2$ and therefore that ...
0
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1answer
33 views

If $d$ is a singular value of an operator $T$, then is $d^2$ a singular value of $T^2$?

I'm trying to prove/disprove a homework problem that is the title question. I'm not looking for an explicit answer, just some direction. So, I've been reading Axler's book Linear Algebra Done Right ...
0
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1answer
63 views

Matrix multiplication/subtraction problem

I have the formula (part of the formulation for a homography): $R*(n^T*x_p*I-t*n^T)$ and I don't understand how it can work given: $R$ is size 3x3 $n^T$ is size 3x1 $x_p$ is size 3x1 $I$ is size ...
0
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1answer
36 views

Calculate total number of matrices of all orders which contain $2013$ elements

Calculate total number of matrices of all orders which contain $2013$ elements My Try:: By Simple Guessing wecan say that there are two matrices of order $(1\times 2013)$ and $(2013 \times 1)$ ...
3
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1answer
518 views

approximating diagonal of inverse sum of low rank and diagonal matrices

I was wondering if there is any theorem or algorithm to approximate the diagonal elements of the inverse of sum of low rank symmetric positive semi-definite and non-negative diagonal matrix. Let me ...
0
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1answer
61 views

Find two matrices such that the sum of their ranks is the rank of the sum

Find $2$ non-zero, $2 \times 2$ matrices, such that $\mathrm{rank}(A+B)=\mathrm{rank}(A)+\mathrm{rank}(B)$ I want to start from the identity matrix and work backwards, but I cant seem to cook up two ...
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0answers
86 views

Minimum absolute determinant of a regular latin square matrix

It is easy to show that a latin square of size n x n has a determinant, which is a multiple of $\large \frac{n^2(n+1)}{2}$, if n is odd and $\large \frac{n^2(n+1)}{4}$, if n is even. This is a lower ...
0
votes
1answer
78 views

Geometric interpretation of an ill-conditioned matrix

Given a non-singular matrix $A\in$ $\Bbb R^{n\times n}$ (invertible) with SVD decomposition $(U, \Sigma, V)$, how would you interpret geometrically $A$ being ill conditioned? From what I know, $A$ is ...
0
votes
1answer
53 views

Commuting matrices

I'm trying to construct an inverse of an operator and it relies on whether I can prove if two matrices commute. Given a matrix function: $A:[0,1] \rightarrow \mathbb{R}^{n \times n}$, I'm trying to ...
1
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2answers
37 views

Matrix theory: Solution of a system

$ A = \left[ {\begin{array}{cc} 1 & 2 & 3 \\ 3 & 7 & 7 \\ 1 & 1 & 3 \end{array} } \right] $ $ B = \left[ {\begin{array}{cc} 1 & 0 & 3 \\ 2 & 1 & 7 \\ 3 & 2 ...
3
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2answers
81 views

Matrix Theory: Orthogonal Matrix

Consider a vector    $$ \begin{array}{l} a=(1,-3,0,1), \\ b=(1,2,0,-1), \\ c=(0,0,1,0) \end{array} $$ Question: (a) A non-zero vector $u$ orthogonal to all three of $a, b, c$ (there ...
1
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1answer
32 views

Rank problem in inversion of t(A) %*% A in R

I need to get the inverse of the cross-product $(\mathbf{A}' \mathbf{A})$, and I run into numerical problems that don't make any sense to me. I actually need $(\mathbf{A}' \mathbf{W}^{-1} ...
2
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0answers
81 views

Strange phenomena in determinants of matrix of determinants.

In my research, my computations are giving rise to the following strange phenomena: Let $$D=\begin{bmatrix}x_1^p & x_2^p & x_{3}^p\\ x_{1}^q & x_{2}^q & x_{3}^q\\ x_{1}^r & ...
2
votes
1answer
84 views

Gauss Seidel Method - How do I avoid calculating $L^{-1}$?

I'm trying to write a matlab code that gets a diagonal dominant matrix $A$, vector $b$, and finds an approximate solution $x$ to $Ax=b$ using Gauss-Seidel Method. I understand the theory. Suppose ...
1
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1answer
53 views

Find the rank of a matrix

I have the matrix $G=(I_{16},A)$ where $A=\left( \begin{array}_ J&I_4&I_4&I_4 \\ I_4&J&I_4&I_4 \\ I_4&I_4&J&I_4 \\ I_4&I_4&I_4&J \end{array} \right)$ ...
2
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0answers
39 views

Square Root of matrices and multiplication

In David Tong's Quantum field theory lecture notes, page 101, line 5, he shows that: $$(p.\sigma)(p.\overline{\sigma})=m^2.I_2$$ (I have placed the identity matrix $I_2$ for clarity) Then I don't ...
18
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4answers
1k views

Why is the trace of a matrix the sum along its diagonal?

Define the trace of a matrix with entries in $\mathbb C$ to be the sum of its eigenvalues, counted with multiplicity. It is a standard (but I think extremely surprising) fact that this is the sum of ...
1
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1answer
100 views

A matrix containing a line such as this is invalid, right? $(0 \space0\space 0\space 0\space|\space1)$

A matrix containing a line such as this is invalid, right? $$ (0 \space0\space 0\space 0\space|\space1) $$ the matrix in question is this: $$\left( \begin{array}{rrrr|r} 1&0&1&1&2\\ ...
3
votes
4answers
141 views

Looking for an elegant proof of $\det(A) = \det(A^t)$ without Schur decomposition

Looking for an elegant proof of $\det(\textbf{A}) = \det(\textbf{A}^{t})$ without Schur decomposition. Proof 1 with Schur decomposition $$\textbf{A} = \textbf{P}^{t}\Delta\textbf{P} ...